Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

History

Metrizable topologies on vector spaces have been studied since their introduction in Maurice Frechet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914, although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).

Electric Universe

Electric Universe is a psychedelic trance project from Germany formed by Boris Blenn and Michael Dressler in 1991. Their first EP release, Solar Energy was an instant hit with the underground trance scene and is often credited with putting the Spirit Zone Recordings label at the forefront of psychedelic trance early on. According to The Sofia Echo, they were "hailed in the 1990s as one of the top psychedelic trance projects to come out of Germany".

History

The Electric Universe project was founded in 1991 by Boris Blenn and Michael Dressler in Hamburg, Germany. After being inspired by the first big Voov Experience in Sprötze, the first Psy Trance orientated tracks were produced, with just 5 pieces of equipment. These were two synthesizers, one sampler, a mixing desk and an Atari 1080. Some of the tracks found their way into the hands of DJ Antaro, who had just started his record label Spirit Zone. He liked the stuff and decided to release the Electric Universe Solar Energy maxi single as the second release on his label. It turned out to be a big hit and set the ground for the first album One Love in 1994.

Stanford University

Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California, and one of the world's most prestigious institutions.

Stanford was founded in 1885 by Leland Stanford, former Governor of and U.S. Senator from California and leading railroad tycoon, and his wife, Jane Lathrop Stanford, in memory of their only child, Leland Stanford, Jr., who had died of typhoid fever at age 15 the previous year. Stanford admitted its first students on October 1, 1891 as a coeducational and non-denominational institution. Tuition was free until 1920. The university struggled financially after Leland Stanford's 1893 death and again after much of the campus was damaged by the 1906 San Francisco earthquake. Following World War II, Provost Frederick Terman supported faculty and graduates' entrepreneurialism to build self-sufficient local industry in what would later be known as Silicon Valley. By 1970, Stanford was home to a linear accelerator, and was one of the original four ARPANET nodes (precursor to the Internet).

Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

Formal definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compact neighbourhood.

There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general:

Logical relations among the conditions:

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.